The Weitzenböck Machine

The Weitzenböck Machine Uwe Semmelmann & Gregor Weingart February 19, 2012 Abstract Weitzenböck formulas are an important tool in relating local differential geometry to global topological properties by means of the so called Bochner method. In this article we give a unified treatment of the construction of all possible Weitzenböck formulas for all irreducible, non symmetric holonomy groups. The resulting classification is two fold, we construct explicitly a basis of the space of Weitzenböck formulas on the one hand and characterize Weitzenböck formulas as eigenvectors for an explicitly known matrix on the other. Both classifications allow us to find tailor suit Weitzenböck formulas for applications like eigenvalue estimates or Betti number estimates. Contents 1 Introduction 1 2 The Holonomy Representation 3 3 The Space W(V ) of Weitzenböck Formulas 8 3.1 Weitzenböck Formulas............................ 9 3.2 The conformal weight operator....................... 11 3.3 The Classifying Endomorphism....................... 16 4 The Recursion Procedure for SO(n), G 2 and Spin(7) 19 4.1 The basic recursion procedure........................ 19 4.2 Computation of B eigenvalues for SO(n), G 2 and Spin(7)........ 21 4.3 Basic Weitzenböck formulas for SO(n), G 2 and Spin(7)......... 22 5 The Weitzenböck Machine for Kähler Holonomies 24 6 A Matrix Presentation of the Twist Operator τ 29 7 Examples 31 8 Bochner Identities in G 2 and Spin(7) Holonomy 37 8.1 Universal Weitzenböck Classes and the Kostant Theorem......... 37 8.2 Proof of the Bochner Identities in Holonomy g 2 and spin 7......... 41 1

A Geometric Proof of the Bicommutant Theorem 43 B Module Generators and Higher Casimirs 45 1 Introduction Weitzenböck formulas are an important tool for linking differential geometry and topology of compact Riemannian manifolds. They feature prominently in the Bochner method, where they are used to prove the vanishing of Betti numbers under suitable curvature assumptions or the non existence of metrics of positive scalar curvature on spin manifolds with non vanishing  genus. Moreover they are used to proof eigenvalue estimates for Laplace and Dirac type operators. In these applications one tries to find a (positive) linear combination of hermitean squares D D of first order differential operators D, which sums to an expression in the curvature only. In this approach one need only consider special first order differential operators D known as generalized gradients or Stein Weiss operators, which are defined as projections of a covariant derivative. Examples for generalized gradients include the exterior derivative d and its adjoint d and the Dirac and twistor operator in spin geometry. In this article we present two different classifications of all possible linear combinations of hermitean squares D D of generalized gradients D, which sum to pure curvature expressions, if the underlying connection is the Levi Civita connection of a Riemannian manifold M of reduced holonomy SO(n), U(n), SU(n) or the exceptional holonomies G 2 and Spin(7). Both classifications are interesting in their own right, the first describes a recursive procedure to calculate a generating set of Weitzenböck formulas, the second classification provides a simple means to decide, whether a given linear combination of hermitean squares of Stein Weiss operators is actually a pure curvature expression. In order to describe the setup of the article in more detail we recall that every representation G Aut V of the holonomy group G of a Riemannian manifold (M, g) on a complex vector space V defines a complex vector bundle V M on M with a covariant derivative induced from the Levi Civita connection, in particular the complexified holonomy representation T of G defines the complexified tangent bundle T M. The generalized gradients on V M are the parallel first order differential operators T ε defined as the projection of : V M T M V M to the parallel subbundles V ε M T M V M arising from a decomposition T V = ε V ε into irreducible subspaces. It will be convenient in this article to call every (finite) linear combination ε c εtε T ε of hermitean squares of generalized gradients a Weitzenböck formula. Our first important observation is that the space W(V ) of all Weitzenböck formulas on a vector bundle V M can be identified with the vector space End g (T V ) and thus is an algebra, which is commutative for irreducible representations V. Moreover it is easy to see that the algebra W(V ) has a canonical involution, the twist τ : W(V ) W(V ), such that a Weitzenböck formula reduces to a pure curvature expression if and only if it is an eigenvector of τ of eigenvalue 1. Of course there are interesting Weitzenböck formulas, which are eigenvectors of τ for the eigenvalue +1, perhaps the most prominent example is the connection Laplacian. The classical examples of Weitzenböck formulas like = d d + d d = + q(r) 2

the original Weitzenböck formula or the Lichnerowicz Weitzenböck formula D 2 = + κ 4 reduce in this setting to the statements that and D 2 respectively are eigenvectors of τ for the eigenvalue 1 and thus pure curvature expressions. Starting with the connection Laplacian, corresponding to 1 End g (T V ), we will describe a recursion procedure to construct a basis of the space W( ) of Weitzenböck formulas on an irreducible vector bundle M on M such that the base vectors are eigenvectors of τ with alternating eigenvalues ±1. Interestingly this recursive procedure makes essential use of a second fundamental Weitzenböck formula B W(V ), the so called conformal weight operator, which was considered for the first time in the work of Paul Gauduchon on conformal geometry [G91]. The details of this recursion procedure and the first initial elements are discussed for the holonomies SO(n), G 2 and Spin(7) only, because the discussion of Weitzenböck formulas in the Kähler holonomies U(n) and SU(n) is better done differently, whereas the hyperkähler holonomies Sp(1) Sp(n) and Sp(n) will be discussed in more detail in a forthcoming paper. Eventually we obtain a sequence of B polynomials p i (B) such that p 2i (B) is in the (+1) and p 2i 1 (B) is in the ( 1) eigenspace of τ. If b ε are the B-eigenvalues on V ε T V, then the coefficient of T ε T ε in the Weitzenböck formula corresponding to p i (B) is given by p i (b ε ). An interesting feature appears for holonomy G 2 and Spin(7). Here we have the decomposition Hom g (Λ 2 T, End V ) = Hom g (g, End V ) Hom g (g, End V ) and because of the holonomy reduction any Weitzenböck formula in the second summand has a zero curvature term. Finally we would like to mention that the problem of finding all possible Weitzenböck formulas is also considered in the work of Y. Homma (e.g. in [H06]). He gives a solution in the case of Riemannian, Kählerian and HyperKählerian manifolds. Even if there are some similarities in the results, it seems fair to say that our method is completely different. In particular we describe an recursive procedure for obtaining the coefficients of Weitzenböck formulas. The main difference is of course that we give a unified approach including the case of exceptional holonomies. 2 The Holonomy Representation For the rest of this article we will essentially restrict to irreducible non symmetric holonomy algebras g. Most of the statements easily generalize to holonomy algebras g with no symmetric irreducible factor in their local de Rham decomposition, which could be called properly non symmetric holonomy algebras. Some of the concepts introduced are certainly interesting for symmetric holonomy algebras as well, in particular the central idea used to find the matrix of the twist through the Recursion Formula 4.1. Turning to 3

irreducible non symmetric holonomy algebras g leaves us with seven different cases algebra g R holonomy representation T R T general Riemannian so n defining representation R n T Kähler u n = ir sun defining representation C n Ē E Calabi Yau su n defining representation C n Ē E quaternionic Kähler sp(1) sp(n) representation H 1 H H n H E hyper Kähler sp(n) defining representation H n C 2 E exceptional G 2 g 2 standard representation R 7 [7] exceptional Spin(7) spin 7 spinor representation R 8 [8] (2.1) according to a theorem of Berger, where T denotes the complexified holonomy representation T := T R R C endowed with the C bilinear extension, of the scalar product. For simplicity we will work with the complexified holonomy representation T and the complexified holonomy algebra g := g R R C throughout as well as with irreducible complex representations of g of highest weight λ. Notations like E or H in the table above fix nomenclature for particularly important representations in special holonomy, say E and Ē refer to the spaces of (1, 0) and (0, 1) vectors in T in the Kähler and Calabi Yau case, while [7] and [8] are the standard 7 dimensional representation of G 2 and 8 dimensional spinor representation of Spin(7) respectively. In passing we note that the complexified holonomy representation T is not isotypical in the Kähler and the Calabi Yau case and this is precisely the reason why these two cases differ significantly from the rest. In order to understand Weitzenböck formulas or parallel second order differential operators it is a good idea to start with parallel first order differential operators usually called generalized gradients or Stein Weiss operators. Their representation theoretic background is the decomposition of tensor products T V of the holonomy representation T with an arbitrary complex representation V. The general case immediately reduces to studying irreducible representations V = of highest weight λ. In this section we will see that the isotypical components of T are always irreducible for a properly non symmetric holonomy algebra g and isomorphic to irreducible representations +ε of highest weight λ + ε for some weight ε of the holonomy representation T. Thus the decomposition of T is completely described by the subset of relevant weights ε: Definition 2.1 (Relevant Weights) A weight ε of the holonomy representation T is called relevant for an irreducible representation of highest weight λ if the irreducible representation +ε of highest weight λ + ε occurs in the tensor product T. We will write ε λ for a relevant weight ε for a given irreducible representation. Lemma 2.2 (Characterization of Relevant Weights) Consider the holonomy representation T of an irreducible non symmetric holonomy algebra g and an irreducible representation of highest weight λ. The decomposition of the tensor product T is multiplicity free in the sense that all irreducible subspaces are pairwise non isomorphic. The complete decomposition of T is thus the sum T = 4 ε λ +ε

over all relevant weights ε. A weight ε 0 is relevant if and only if λ + ε is dominant. The zero weight ε = 0 only occurs for the holonomy algebras so n with n odd and g 2, it is relevant if λ λ Σ or λ λ T respectively is still dominant, where λ Σ and λ T are the highest weights of the spinor representation of so n and the standard representation of g 2. Proof: The proof is essentially an exercise in Weyl s character formula A particular consequence of Lemma 2.2 is that for sufficiently complicated representations all weights ε of the holonomy representation T are relevant. With this motivation we will call a highest weight λ generic if λ + ε is dominant for all weights ε of the holonomy representation T. A simple consideration shows that λ is generic if and only if λ ρ is dominant, where ρ is the half sum of positive roots or equivalently the sum of fundamental weights, unless we consider odd dimensional generic holonomy g = so 2r+1 or g = g 2. In the latter holonomies the generic weights λ must have λ ρ λ Σ or λ ρ λ T dominant respectively. In any case the number of relevant weights for the representation N(G, λ) := { ε ε is relevant for λ } dim T is bounded above by dim T with equality if and only if λ is generic. In particular there are at most dim T summands in the decomposition of T into irreducibles, exactly one copy of +ε for every relevant weight ε. On the other hand the number N(G, λ) of irreducible summands in the decomposition of T agrees with the dimension of the algebra End g ( T ) of g invariant endomorphisms of T, because all isotypical components are irreducible by Lemma 2.2. In the next section we will study the identification End g ( T ) = Hom g ( T T, End ) extensively, which allows us to break up End g ( T ) into interesting subspaces called Weitzenböck classes, whose dimension can be calculated in the following way: Lemma 2.3 (Dimension of Weitzenböck Classes) Let us call the space W t := Hom t (R, W ) W of elements of a G representation W invariant under a fixed Cartan subalgebra t g the zero weight space of W. The dimension of the zero weight space provides an upper bound dim Hom g ( W, End ) dim W t for the dimension of the space Hom g ( W, End ) for an irreducible representation. For sufficiently dominant highest weight λ in dependence on W this upper bound is sharp. The lemma follows again from the Weyl character formula, but it is also an elementary consequence of Kostant s theorem 8.3 formulated below. We will mainly use Lemma 2.3 for the subspaces W α occuring in the decomposition T T = W α into irreducibles. In the case of the holonomy algebras so n, g 2 and spin 7 we have the decomposition T T = C Sym 2 0T g g and the following dimensions of the zero weight spaces: dim T dim [ C ] t dim [ Sym 2 0T ] t dim [ g ] t dim [ g ] t so n n 1 n 1 2 n 2 g 2 7 1 2 3 1 spin 7 8 1 3 3 1 (2.2) 5

Note in particular that the dimensions of the zero weight spaces sum up to dim T. Although complete the decision criterion given in Lemma 2.2 is not particularly straightforward in general. At the end of this section we want to give a graphic interpretation of this decision criterion for all irreducible holonomy groups in order to simplify the task of finding the relevant weights. For a fixed holonomy algebra g the information necessary in this graphic algorithm is encoded in a single diagram featuring the weights of the holonomy representation T and labeled boxes. A weight ε is relevant for an irreducible representation if and only if the highest weight λ = λ 1 ω 1 +... + λ r ω r of, written as a linear combination of fundamental weights ω 1,..., ω r, satisfies all inequalities labeling the boxes containing ε. The notation introduced for the weights of the holonomy representation T and the fundamental weights will be used throughout this article. To begin with let us consider even dimensional Riemannian geometry with generic holonomy g = so 2r, r 2. In this case the holonomy representation T is the defining representation, whose weights ±ε 1, ±ε 2,..., ±ε r form an orthonormal basis for a suitable scalar product, on the dual t of the maximal torus. The ordering of weights can be chosen in such a way that the fundamental weights ω 1,..., ω r are given by: ω 1 = ε 1 ±ε 1 = ± ω 1 ω 2 = ε 1 + ε 2 ±ε 2 = ±(ω 2 ω 1 ).... ω r 2 = ε 1 +... + ε r 2 ±ε r 2 = ±(ω r 2 ω r 3 ) ω r 1 = 1(ε 2 1 +... + ε r 1 + ε r ) ±ε r 1 = ±(ω r 1 + ω r ω r 2 ) ω r = 1(ε 2 1 +... + ε r 1 ε r ) ±ε r = ±(ω r 1 ω r ) Every dominant integral weight of so 2r can be written λ = λ 1 ω 1 +...+λ r ω r with natural numbers λ 1,..., λ r 0 and the criterion of Lemma 2.2 becomes: +ε 1 λ 1 1 ε 1 +ε 2 λ 2 1 ε 2 +ε 3 λ r 2 1 ε r 2 +ε r 1 λ r 1 ε r 1 ε r +ε r A weight ε of the holonomy representation T of so 2r is relevant for the irreducible representation if and only if λ satisfies all the conditions labeling the boxes containing ε. Say the weights ε 1 and +ε 2 are relevant for all irreducible representations with λ 1 1, whereas ε r 1 is relevant for if and only if λ r 1 1 and λ r 1. Odd dimensional Riemannian geometry g = so 2r+1, r 1, with generic holonomy is of course closely related to g = so 2r. The weights ±ε 1, ±ε 2,..., ±ε r of the defining holonomy representation T besides the zero weight form an orthonormal basis for a suitable scalar product, on the dual t of the maximal torus. With a suitable choice of 6

ordering of weights the fundamental weights ω 1,..., ω r and the weights of T relate via: ω 1 = ε 1 ±ε 1 = ± ω 1 ω 2 = ε 1 + ε 2 ±ε 2 = ±(ω 2 ω 1 ).... ω r 1 = ε 1 +... + ε r 1 ±ε r 2 = ±(ω r 1 ω r 2 ) ω r = 1(ε 2 1 +... + ε r 1 + ε r ) ±ε r = ±(2ω r ω r 1 ) Writing a dominant integral weight λ = λ 1 ω 1 +... + λ r ω r as a linear combination of fundamental weights with integers λ 1,..., λ r 0 the criterion of Lemma 2.2 becomes: +ε 1 λ 1 1 ε 1 +ε 2 λ 2 1 ε 2 +ε 3 λ r 2 1 ε r 2 +ε r 1 λ r 1 1 ε r 1 +ε r λ r 1 ε r 0 λ r 2 Turning from the Riemannian case to the Kähler case g = u n we observe that the weights ±ε 1,..., ±ε n of the defining standard representation T = E Ē form an orthonormal basis for an invariant scalar product on the dual t of a maximal torus t u n, but they become linearly dependent when projected to the dual of a maximal torus of the ideal su n u n. In any case the fundamental weights and the weights of T relate as ω 1 = ε 1 ±ε 1 = ± ω 1 ω 2 = ε 1 + ε 2 ±ε 2 = ±(ω 2 ω 1 ).... ω n = ε 1 +... + ε n ±ε n = ±(ω n ω n 1 ) and the criterion of Lemma 2.2 becomes: +ε 1 λ 1 1 ε 1 +ε 2 λ 2 1 ε 2 +ε 3 λ n 2 1 ε n 2 +ε n 1 λ n 1 1 ε n 1 +ε n ε n The quaternionic Kähler and hyperkähler cases are more complicated, because the condition of being relevant has to be checked for both ideals sp(1) and sp(n) of g. For a single 7

ideal however the condition becomes simple again. The weights ±ε 1,..., ±ε n of E are again orthonormal for a suitable scalar product, on the dual t of a maximal torus t sp(r) for r = 1 or r = n and relate to the fundamental weights by the formulas: ω 1 = ε 1 ±ε 1 = ± ω 1 ω 2 = ε 1 + ε 2 ±ε 2 = ±(ω 2 ω 1 ).... ω r = ε 1 +... + ε r ±ε r = ±(ω r ω r 1 ) The graphical interpretation of Lemma 2.2 is given by the diagram: +ε 1 λ 1 1 ε 1 +ε 2 λ 2 1 ε 2 +ε 3 λ r 2 1 ε r 2 +ε r 1 λ r 1 1 ε r 1 +ε r λ r 1 ε r Finally we consider the two exceptional cases g 2 and spin 7. Recall that the group G 2 is the group of automorphisms of the octonions O as an algebra over R. In this sense the holonomy representation T R is the defining representation Im O of g 2 with complexification T = [7]. There are too many weights of the holonomy representation to be orthonormal for any scalar product on the dual t of a fixed maximal torus t g 2, but at least we can choose an ordering of weights for t so that the weights of T become totally ordered +ε 1 > +ε 2 > +ε 3 > 0 > ε 3 > ε 2 > ε 1. In this notation we have: ω 1 = ε 1 ±ε 1 = ± ω 1 ω 2 = ε 1 + ε 2 ±ε 2 = ±(ω 2 ω 1 ) ±ε 3 = (ω 2 2ω 1 ) The scalar product of choice on t is specified by ε 1, ε 1 = 1 = ε 2, ε 2 and ε 1, ε 2 = 1 2. Writing a dominant integral weight as λ = aω 1 + bω 2, a, b 0, we read Lemma 2.2 as: +ε 1 a 1 ε 1 +ε 2 0 b 1 ε 2 +ε 3 a 2 ε 3 The holonomy representation of the holonomy algebra g = spin 7 is the 8 dimensional spinor representation T = [8]. It is convenient to write the weights ±ε 1,..., ±ε 4 of T 8

and the fundamental weights ω 1, ω 2 and ω 3 in terms of the weights ±η 1, ±η 2, ±η 3, 0 of the 7 dimensional defining representation of spin 7, which form an orthonormal basis for a suitable scalar product on the dual t of the maximal torus. With this proviso the weights ±ε 1,..., ±ε 4 of T and the fundamental weights ω 1, ω 2 and ω 3 can be written as: ω 1 = η 1 ±ε 1 = ± 1(η 2 1 + η 2 + η 3 ) = ± ω 3 ω 2 = η 1 + η 2 ±ε 2 = ± 1(η 2 1 + η 2 η 3 ) = ±(ω 2 ω 3 ) ω 3 = 1 2 (η 1 + η 2 + η 3 ) ±ε 3 = ± 1 2 (η 1 η 2 + η 3 ) = ±(ω 3 ω 2 + ω 1 ) ±ε 4 = ± 1 2 (η 1 η 2 η 3 ) = ±(ω 1 ω 3 ) and Lemma 2.2 for a dominant integral weight λ = aω 1 + bω 2 + cω 3 translates into: +ε 1 c 1 ε 1 +ε 2 b 1 ε 2 a 1 ε 4 ε 3 +ε 4 +ε 3 3 The Space W(V ) of Weitzenböck Formulas In this section we will define twistor operators, Weitzenböck formulas and the space of Weitzenböck formula with its different realizations. Then we will introduce the conformal weight operator, which in many cases generates all possible Weitzenböck formulas. Finally we define the classifying endomorphism and study the corresponding eigenspace decomposition. 3.1 Weitzenböck Formulas We consider parallel second order differential operators P on sections of a vector bundle V M over a Riemannian manifold M with special holonomy G. By definition these are differential operators, which up to first order differential operators can always be written as the composition Γ(V M) 2 Γ(T M T M V M) = Γ(T M T M V M) F Γ(V M) where F is a parallel section of the vector bundle Hom (T M T M V M, V M) corresponding to a G equivariant homomorphism F Hom G (T T V, V ). A particularly important example is the connection Laplacian which arises from the linear map a b ψ a, b ψ. Note that we are only considering reduced holonomy groups G, which are connected by definition, so that G equivariance is equivalent to g equivariance. Taking advantage of this fact we describe other parallel differential operators by means of the following identifications of spaces of invariant homomorphisms: Hom g ( T T V, V ) = Hom g ( T T, End V ) = End g ( T V ) 9

Of course the identification Hom g (T T V, V ) = Hom g (T T, End V ) is the usual tensor shuffling F (a b v) = F a b v for all a, b T and v V. The second identification Hom g (T T V, V ) = End g (T V ) depends on the existence of a g invariant scalar product on T or the musical isomorphism T = T via a summation F ( b v ) = µ t µ F (t µ b v) F (a b v) = ( a, id) F (b v) over an orthonormal basis {t µ }. Under this identification the identity of T V becomes the homomorphism a b ψ a, b ψ corresponding to the connection Laplacian. The composition of endomorphisms turns End g ( T V ) and thus Hom g (T T, End V ) into an algebra, for F, F Hom g (T T, End V ) the resulting algebra structure reads: ( F F ) a b = µ F a tµ F tµ b (3.3) Last but not least we note that the invariance condition for F Hom g ( T T, End V ) is equivalent to the identity [ X, F a b ] = F Xa b + F a Xb for all X g and a, b T. Assuming that V = is irreducible of highest weight λ we know from Lemma 2.2 that the isotypical components of T are irreducible for non symmetric holonomy groups. The algebra End g ( T ) is thus commutative and spanned by the pairwise orthogonal idempotents pr ε projecting onto the irreducible subspaces +ε of T. In order to describe the corresponding second order differential operators we introduce first order differential operators T ε known as Stein Weiss operators or generalized gradients by: T ε : Γ( M ) Γ( +ε M ), ψ pr ε ( ψ). A typical example of a Stein Weiss operator is the twistor operator of spin geometry, which projects the covariant derivative of a spinor onto the kernel of the Clifford multiplication. Straightforward calculations show that the second order differential operator associated to the idempotent pr ε is the composition of T ε with its formal abjoint operator T ε : Γ(+ε M) Γ( M) in the sense pr ε ( 2 ) = T ε T ε compare [S06]. In consequence we can write the second order differential operator F ( 2 ) associated to F Hom g ( T T, ) as a linear combination of the squares of Stein Weiss operators: F ( 2 ) = ε f ε T ε T ε. (3.4) In fact with End g ( T ) = Hom g ( T T, End ) being spanned by the idempotents pr ε every F End g ( T ) expands as F = ε f ε pr ε with coefficients f ε determined by F Vλ+ε = f ε id. A particular instance of equation (3.4) is the identity = ε T ε T ε associated to the expansion id T Vλ = ε pr ε. Motivated by this and other well known identities of second order differential operators of the form (3.4) we will in general call all elements F Hom g ( T T V, V ) = End g (T V ) Weitzenböck formulas: Definition 3.1 (Space of Weitzenböck Formulas on V M) The Weitzenböck formulas on a vector bundle V M correspond bijectively to vectors in: W( V ) := Hom g ( T T V, V ) = Hom g ( T T, End V ) = End g ( T V ). 10

Of course we are mainly interested in Weitzenböck formulas inducing differential operators of zeroth order or equivalently pure curvature terms. Clearly a Weitzenböck formula F Hom g ( T T V, V ) skew symmetric in its two T arguments will induce a pure curvature term F ( 2 ), because we can then reshuffle the summation in the calculation: F ( 2 v ) = 1 F ( t 2 µ t ν ( 2 t µ,t ν 2 t ν,t µ )v ) = 1 F 2 tµ tν R tµ, t ν v. (3.5) µν Here and in the following we will denote with {t ν } an orthonormal basis of T and also a local orthonormal basis of the tangent bundle. Conversely the principal symbol of the differential operator F ( 2 ) is easily computed to be σ F ( 2 )(ξ)v = F ξ ξ v for every cotangent vector ξ and every v V M. Hence the principal symbol vanishes identically exactly for the skew symmetric Weitzenböck formulas. Weitzenböck formulas F leading to a pure curvature term F ( 2 ) are thus completely characterized by being eigenvectors of eigenvalue 1 for the involution τ : W( V ) W( V ), F τ( F ) defined in the interpretation W(V ) = Hom g ( T T V, V ) as precomposition with the twist τ : T T V T T V, a b v b a v. In other words a Weitzenböck formula F will reduce to a pure curvature term if and only if τ(f ) := F τ = F. Considering the space of Weitzenböck formulas W(V ) as the algebra End g (T V ) we can introduce additional structures on it: the unit 1 := id T V W(V ), the scalar product F, F := 1 dim V tr T V ( F F ) F, F W(V ) satisfying F G, F = F, G F and the trace tr F := F, 1. Clearly the trace of the unit is given by tr 1 = dim T. The definition of the trace can be rewritten in the form 1 ( tr F = dim V tr V v ) F tµ t µ v µ so that the trace is invariant under the twist τ. A slightly more complicated argument using (3.3) shows that the scalar product is invariant under the twist, too. In particular the eigenspaces for τ for the eigenvalues ±1 are orthogonal and all eigenvectors in the ( 1) eigenspace of τ have vanishing trace. Fom the definition of the trace we obtain that the trace of an element F = f ε pr ε of W( ) in the irreducible case is given by µν tr F = ε f ε dim +ε dim (3.6) in particular the idempotents pr ε form an orthogonal basis of W( ): pr ε, pr ε = δ ε ε dim +ε dim A different way to interprete the trace is to note that for every Weitzenböck formula F W(V ) considered as an equivariant homomorphism F : T T End V the trace endomorphism µ F t µ t µ End g V is invariant under the action of g. For an irreducible representation it is thus the multiple µ F t µ t µ = (tr F ) id Vλ of id Vλ by Schur s Lemma. 11

3.2 The conformal weight operator In order to study the fine structure of the algebra W(V ) = End g (T V ) of Weitzenböck formulas it is convenient to introduce the conformal weight operator B W(V ) of the holonomy algebra g and its variations B h W(V ) associated to the non trivial ideals h g of g. All these conformal weight operators commute and the commutative subalgebra of W(V ) generated by them in the irreducible case V = is actually all of W(V ) for generic highest weight λ. In this subsection we work out some direct consequences of the description of Weitzenböck formulas as polynomials in the conformal weight operators. Recall that the scalar product, on T induces a scalar product on all exterior powers Λ k T of T via Gram s determinant. Using this scalar product on Λ 2 T we can identify the adjoint representation so T of SO(n) with Λ 2 T through X, a b = Xa, b and hence think of the holonomy algebra g so T as a subspace of the euclidian vector space Λ 2 T : Definition 3.2 (Conformal Weight Operator) Consider an ideal h R g R in the real holonomy algebra. Its complexification h := h R R C is an ideal in g and a regular subspace h g Λ 2 T in Λ 2 T with associated orthogonal projection pr h : Λ 2 T h. The conformal weight operator B h W(V ) is defined by B h a b v := pr h(a b) v in the interpretation of Weitzenböck formulas as linear maps T T End V. Under the identification Hom g (T T, End V ) = End g (T V ) discussed above the conformal weight operator B h becomes the following sum over an orthonormal basis {t µ } of T : B h ( b v ) = µ t µ pr h (t µ b) v The notation B := B g will be used for the conformal weight operator of the algebra g. Most of the irreducible non symmetric holonomy algebras g are simple and hence there is only one weight operator B ( c.f. table (2.1)). The exceptions are Kähler geometry g R = ir su n with a one dimensional center in dimension 2n and two commuting weight operators B ir and B su and quaternionic Kähler geometry g R = sp(1) sp(n) in dimension 4n, n 2 with two commuting weight operators B H and B E. Lemma 3.3 (Fegan s Lemma [F76]) The conformal weight operator B h W(V ) of an ideal h g Λ 2 T can be written B h = α X α X α End g ( T V ) where {X α } is an orthonormal basis of h for the scalar product on Λ 2 T induced from T. Proof: Let {t µ } and {X α } be orthonormal bases of T and h respectively. Using the characterization X, a b = Xa, b of the identification so T = Λ 2 T we find: B h ( b v ) = µ t µ pr h ( t µ b ) v = µα t µ X α, t µ b X α v = α X α b X α v. 12

A particularly nice consequence of Fegan s Lemma is that the conformal weight operators B h and B h associated to two ideals h, h g always commute. In fact two disjoint ideals h h = {0} of g commute by definition, the general case follows easily. Hence the algebra structure on W(V ) allows us to use the evaluation homomorphism Φ : C[ { B h h irreducible ideal of g } ] W( V ) (3.7) from the polynomial algebra on abstract symbols {B h } to the algebra End g (T V ) for studying the fine structure of the space W(V ) of Weitzenböck formulas. In order to turn Fegan s Lemma into an effective formula for the eigenvalues of the conformal weight operator B h of an ideal h g we need to calculate the Casimir operator in the normalization given by the scalar product on Λ 2 T. Recall that the Casimir operator is defined as a sum Cas := α X2 α U h over an orthonormal basis {X α } of h and is thus determined only up to a constant. Usually it is much more convenient to calculate the Casimir Cas with respect to a scalar product of choice and later normalize it to the Casimir Cas Λ2 with respect to the invariant scalar product induced on h Λ 2 T. For a given irreducible ideal h g in an irreducible holonomy algebra g the Casimir operator Cas for h is now real, symmetric and g invariant. Although the holonomy representation T of g may not be irreducible itself, it is the complexification of the irreducible real representation T R so that we can still conclude that Cas acts as the scalar multiple Cas T id of the identity on T. The Casimir eigenvalue Cas Λ2 of the properly normalized Casimir operator on a general irreducible representation of g of highest weight λ can then be calculated from the Casimir Cas using Cas Λ2 = 2 dim h dim T Cas Vλ Cas T, (3.8) where the ambiguity in the choice of normalization cancels out in the quotient Cas Cas T. In fact the normalization (3.8) is readily checked for the holonomy representation V = T tr T Cas Λ2 = dim T Cas Λ2 T = α tr T X 2 α = 2 dim h, because the scalar product induced from T on Λ 2 T satisfies X, Y = 1 2 tr T XY. Corollary 3.4 (Explicit Formula for Conformal Weights) Consider the tensor product T = ε λ +ε of the holonomy representation T with the irreducible representation of highest weight λ. For an ideal h g let ε max be the highest weight of T and ρ be the half sum of positive weights of h in the dual t of a maximal torus t. With respect to the basis {pr ε } of idempotents the conformal weight operator B h of the ideal h can be expanded B h = ε λ bh ε pr ε with conformal weights b h ε = 2 dim h dim T λ + ρ, ε ρ, ε max + 1 2 ( ε 2 ε max 2 ) ε max + 2ρ, ε max, where, is an arbitrary scalar product on t invariant under the Weyl group of h. 13

Proof: According to Lemma 3.3 the conformal weight operator can be written as a difference B h = 1 2 (CasΛ2 Cas Λ2 id id Cas Λ2 ) of properly normalized Casimir operators. In particular its restriction to the irreducible summand +ε T acts by multiplication with b h ε := 1 2 (CasΛ2 +ε Cas Λ2 T Cas Λ2 ). The conformal weights b h ε can thus be calculated using Freudenthal s formula Cas Vλ = λ + 2ρ, λ for the Casimir eigenvalues of irreducible representations and the normalization (3.8). It is clear from the definition that the conformal weight operator B h W(V ) of an ideal h g of the holonomy algebra g is in the ( 1) eigenspace of the involution τ and thus induces a pure curvature term B h ( 2 ) on every vector bundle V M associated to the holonomy reduction of M. Explicitly we can describe this curvature term using an orthonormal basis {X α } of the ideal h for the scalar product induced on h Λ 2 T. Namely the curvature operator R : Λ 2 T M gm Λ 2 T M, a b R a,b, associated to the curvature tensor R of M allows us to write down a well defined global section q h (R) := α X α R(X α ) Γ( U 2 gm ) (3.9) of the universal enveloping algebra bundle associated to the holonomy reduction. Fixing a representation G Aut(V ) of the holonomy group the section q h (R) in turn induces an endomorphism on the vector bundle V M associated to V and the holonomy reduction. A particularly important example of Weitzenböck formulas is the classical formula of Weitzenböck for the Laplace operator = d d + d d acting on differential forms, i.e. = + q(r) (3.10) The curvature term in this formula is precisely the curvature endomorphism for the full holonomy algebra g, in particular q(r) = Ric on the bundle of 1 forms on M. We recall that the curvature term in the Weitzenböck formula (3.10) is known to be the Casimir operator of the holonomy algebra g on a symmetric space M = G/G, more precisely for every ideal h g the curvature term q h (R) acts as the Casimir operator of the ideal h on every homogeneous vector bundle V M over a symmetric space M. A minor subtlety in the definition of the curvature terms q h (R) should not pass unnoticed, in difference to the conformal weight operators B h the curvature terms q h (R) associated to two ideals h, h g do not in general commute. The problem is that the curvature operator R : Λ 2 T M gm does not necessarily map the parallel subbundle hm Λ 2 T M associated to an ideal h g to itself due to the presence of mixed terms in the curvature tensor R. In other words the section q h (R) is not in general a section of U 2 hm U 2 gm, so it is of no use that disjoint ideals centralize each other. Interestingly the problem of mixed terms is absent for symmetric holonomies as well as for the holonomies so n, g 2, spin 7 and the quaternionic holonomies sp(n) and sp(1) sp(n). Mixed terms may however spoil commutativity of the curvature endomorphisms q h (R) on a Kähler manifold M, in fact the curvature term associated to the center ir u n reads q ir ( R ) = 1 n I ( IRic ) Γ( U 2 um ), according to equation (5.29), where I is the parallel complex structure and IRic is the composition of I with the symmetric Ricci endomorphism of T M thought of as a section 14

of the holonomy bundle um. In consequence q ir (R) is a section of U 2 irm if and only if M is Kähler Einstein, otherwise it will not in general commute with q su (R). The central curvature term features prominently in the Bochner identity for Kähler manifolds. Lemma 3.5 B h ( 2 ) = q h ( R ) Proof: Expanding the second covariant derivative 2 ψ = t µ t ν 2 t µ, t ν ψ of the section ψ with an orthonormal basis {t µ } of T and using the same resummation as in the derivation of equation (3.5) we find for an orthonormal basis {X α } of the ideal h: B h ( 2 ψ ) = 1 2 = 1 2 pr h (t µ t ν ) Rt V µ, t ν ψ µν t µ t ν, X α X α Rt V µ, t ν ψ = q h ( R ) αµν On the other hand Corollary 3.4 tells us how to write the conformal weight operator B h in terms of the basis {pr ε } of projections onto the irreducible summands +ε T. Using the identification of B h ( 2 ) with the universal curvature terms q h ( R ) proved above we obtain some prime examples of Weitzenböck formulas: Proposition 3.6 (Universal Weitzenböck Formula) Consider a Riemannian manifold M of dimension n with holonomy group G SO(n) and the vector bundle M over M associated to the holonomy reduction of M and the irreducible representation of G of highest weight λ. In terms of the Stein Weiss operators T ε : Γ( M) Γ(+ε M) arising from the decomposition T = ε λ +ε the action of the curvature endomorphisms q h (R) can be written q h (R) = ε λ b h ε T ε T ε, where the b h ε are the eigenvalues of the conformal weight operator B h End g (T ). As a direct consequence of Proposition 3.6 and the classical Weitzenböck formula (3.10) for the Laplace operator = dd + d d on the bundle of differential forms we obtain = ε λ (1 b ε ) T ε T ε. In the case of Riemannian holonomy G = SO(n) the universal Weitzenböck formula of Proposition 3.6 was considered in [G91] for the first time. The definition of the conformal weight operator and its expression in terms of the Casimir is taken from the same article. The conformal weight operator B has been used for other purposes as well, see [CGH00] for example. Similar results can be found in [H04]. Considering B as an element of the algebra W(V ) all powers of B are g invariant endomorphisms. In the interpretation W(V ) = Hom g (T T, End V ) these powers read: Ba b k = pr g (a t µ1 ) pr g (t µ1 t µ2 )... pr g (t µk 2 t µk 1 ) pr g (t µk 1 b). (3.11) µ 1,...,µ k 1 15

Recall now that in the irreducible case the trace F tµ t µ = (tr F )id Vλ of an element F W( ) is a multiple of the identity of. Evidently the traces of the powers B k of B correspond to the action of the elements Cas [k] := pr g (t µ0 t µ1 ) pr g (t µ1 t µ2 )... pr g (t µk 2 t µk 1 ) pr g (t µk 1 t µ0 ). (3.12) µ 1,...,µ k 1 of the universal enveloping algebra U g on V. The elements Cas [k], k 2, all belong to the center of the universal enveloping algebra U g and are called higher Casimirs since Cas [2] = 2Cas Λ2 (c.f. [CGH00]). A straightforward calculation shows ( ) Cas [k] = tr (B k ) id Vλ = b k dim +ε ε id Vλ, (3.13) dim for an irreducible representation V =, where we use the second equation in (3.6) for computing the trace of B k = b k εpr ε explicitly. As an example we consider the equation Cas [3] = 1 2 CasΛ2 g Cas Λ2, which follows from the recursion formula of Corollary 4.2 or by direct calculation. Indeed B 2 1 4 CasΛ2 g B is an eigenvector of the involution τ for the eigenvalue +1. Thus it is orthogonal to the eigenvector B for the eigenvalue 1 and so: 0 = B 2 1 4 CasΛ2 g B, B = tr(b 3 ) 1 4 CasΛ2 g tr(b 2 ) = tr(b 3 ) + 1 2 CasΛ2 g Cas Λ2. From a slightly more general point of view the evaluation at the conformal weight operator defines an algebra homomorphism Φ : C[B] End g (T V ), whose kernel is generated by the minimal polynomial of B as an endomorphism on T V. With B being diagonalizable its minimal polynomial is the product min(b) = b {b ε} (B b) over all different conformal weights. In consequence the injective algebra homomorphism Φ : C[B]/ min( B ) End g ( T V ). is an isomorphism as soon as all conformal weights are pairwise different. Indeed the dimension of End g (T V ) is the number N(G, λ) of relevant weights or the number of conformal weights counted with multiplicity, while the number of different conformal weights determines the degree of min(b) and so the dimension of C[B]/ min(b). In section 4.2 we compute the B eigenvalues in the cases g = so n, g 2 and spin 7. It follows that they are pairwise different with the only exceptions of g = so 2r and a representation of highest weight λ = λ 1 ω 1 +...+λ r ω r, with λ r = λ r 1, which is equivalent to b εr = b εr and g = spin 7 and a representation with highest weight λ = aω 1 +bω 2 +cω 3 with c = 2a + 1, which is equivalent to b ε4 = b ε4. In these cases the degree of the minimal polynom is reduced by one and hence the image of Φ has codimension one. We thus have proved the following Proposition 3.7 (Structure of the Algebra of Weitzenböck Formulas) Let G be one of the holonomy groups SO n, G 2 or Spin(7) of non symmetric manifolds. If is irreducible, then Φ is an isomorphism Φ : C[B]/ min( B ) ε = End g ( T ), with the only exception of the cases G = SO 2r and a highest weight λ with λ r 1 = λ r, or G = Spin(7) and a highest weight λ = aω 1 + bω 2 + cω 3 with c = 2a + 1. In both cases the homomorphism Φ is not surjective and its image has codimension one. 16

3.3 The Classifying Endomorphism The decomposition of the space W(V ) = Hom g (T T, End V ) of Weitzenböck formulas into the (±1) eigenspaces of the involution τ can be written as Hom g (T T, End V ) = Hom g (Λ 2 T, End V ) Hom g (Sym 2 T, End V ). However in general we have a further splitting of T T leading to a further decomposition of the τ eigenspaces. Our aim is now to introduce an endomorphism K on W(V ) whose eigenspaces correspond to this finer decomposition. Definition 3.8 (The Classifying Endomorphism) The classifying endomorphism K h of an ideal h R g R of the real holonomy algebra g R is the endomorphism K h : W(V ) W(V ) on the space of Weitzenböck formulas defined in the interpretation W(V ) = Hom g (T T, End V ) by the formula K h ( F ) a b v := α F Xαa X αbv where {X α } is an orthonormal basis for the scalar product induced on the ideal h Λ 2 T. As before we denote the classifying endomorphism of the ideal g simply by K := K g. Of course the definition of the classifying endomorphism K h is motivated by Fegan s Lemma 3.3 for the conformal weight operator B h. Note that for every g equivariant map F : T T End V the map K h (F ) : T T End V is again g equivariant, because we sum over an orthonormal basis {X α } of the ideal h g for a g invariant scalar product. In the interpretation W(V ) = End g (T V ) the definition of K h reads K h ( F )(b v) = µα t µ F Xαt µ X αb v = µα X α t µ F tµ X αb v or more succinctly: K h ( F ) = α ( X α id ) F ( X α id ). (3.14) In consequence the classifying endomorphisms K h and K h for two ideals h, h g commute on the space W(V ) of Weitzenböck formulas similar to the conformal weights operators. The classifying endomorphisms will be extremely useful in finding the matrix of the twist τ : W(V ) W(V ) in the basis of W(V ) given by the orthogonal idempotents pr ε. Lemma 3.9 (Eigenvalues of the Classifying Endomorphism) Consider the decomposition of the tensor product T T = α W α into irreducible summands. The classifying endomorphisms K h are diagonalizable on Hom g (T T, End V ) with eigenspaces Hom g ( W α, End V ) Hom g (T T, End V ) for relevant α and eigenvalues: κ Wα = 1 2 CasΛ2 W α Cas Λ2 T. In particular the classifying endomorphisms K h acts as K h (F ) = α κ W α F Wα on the space of Weitzenböck formulas W(V ) = Hom g (T T, End V ). 17

Proof: From the very definition of K h we see that it acts by precomposition with the map X α X α in the interpretation W(V ) = Hom g (T T, End V ) of the space of Weitzenböck formulas. The argument used in the proof of Corollary 3.4 shows that K h is actually a difference of Casimir operators leading to the stated formula for its eigenspaces and eigenvalues. In the case of the holonomies so n, g 2 and spin 7 we have T T = C Sym 2 0T g g and using Lemma 3.9 we find the following K eigenvalues: κ C κ Sym 2 0 T κ g κ g so n (n 1) 1 1 2 g 2 4 3 0 2 spin 7 21 3 4 4 1 4 9 4 (3.15) Note that all these K eigenvalues are different and consequently the twist τ is a polynomial in the classifying endomorphism K. Moreover a given invariant homomorphism F Hom g (T T, End V ) is an eigenvector of K if and only if F is different from zero one precisely one summand W α T T, so 1 and B are clearly K eigenvectors. Lemma 3.10 (Properties of the Classifying Endomorphism) The classifying endomorphism K : W(V ) W(V ) is a symmetric endomorphism commuting with the twist map τ on the space W(V ) of Weitzenböck formulas equipped with the scalar product F, F := 1 tr dim V T V F F. The special endomorphisms 1 and B for the same ideal h g are K eigenvectors: K( 1 ) = Cas Λ2 T 1 K( B ) = ( Cas Λ2 T 1 2 CasΛ2 h ) B Proof: The symmetry of K is a trivial consequence of equation (3.14) in the form K( F ), F = 1 dim V ( tr T V (X ν id) F (X ν id) F ) ν and the cyclic invariance of the trace, moreover K commutes with τ by definition. Coming to the explicit determination of K(1) and K(B) we observe that the unit of End g (T V ) becomes the equivariant map 1(a b) = a, b id V in Hom g (T T, End V ) and so: ( K1 )(a b) = ν X ν a, X ν b id V = ν a, X 2 ν b id V = Cas Λ2 T 1(a b). The conformal weight operator B considered as an element of Hom g (T T, End V ) lives by definition in the eigenspace Hom g (g, End V ) for the eigenvalue 1 2 CasΛ2 g + Cas Λ2 T of K, where Cas Λ2 g is the Casimir eigenvalue of the adjoint representation. On a manifold with holonomy algebra g so n = Λ 2 T the Riemannian curvature tensor takes values in g, i.e. it can be considered as an element of Sym 2 g. This fact has the following important consequence: 18

Proposition 3.11 (Bochner Identities) Suppose F W(V ) is an invariant homomorphism T T End V factoring through the projection to the orthogonal complement g Λ 2 T T T of g Λ 2 T. The curvature expression F ( 2 ) = 0 vanishes regardless of what the curvature tensor R is. We will call a Weitzenböck formula F W( ) corresponding to an invariant homomorphism T T End V factoring through g a Bochner identity. Writing such an invariant homomorphism F in terms of the basis {pr ε } as F = f ε pr ε we get the following explicit form of the Bochner identity: f ε Tε T ε = 0. ε The Bochner identities of G 2 and Spin(7) holonomy correspond to eigenvectors of the classifying endomorphism K for the eigenvalues 2 and 9 respectively. Since the zero 4 weight space of g is in both cases one-dimensional it follows from Lemma 2.3 that dim Hom g (g, End ) 1, (3.16) i. e. there is at most one Bochner identity. Moreover the K eigenvector 1 End g (T ) spans the K eigenspace Hom g (C, End ) = C. Because the zero weight space of g itself is the fixed Cartan subalgebra t g, an application of Lemma 2.3 results in the estimates: dim Hom g2 (g 2, End ) 2 dim Hom spin7 (spin 7, End ) 3. 4 The Recursion Procedure for SO(n), G 2 and Spin(7) The definitions of the conformal weight operator B and the classifying endomorphism K given in the previous section are very similar. Given this similarity it should not come as a surprise that the actions of B and K on the space W(V ) of Weitzenböck formulas obey a simple relation, which is the corner stone of the treatment of Weitzenböck formulas proposed in this article. In the present section we first prove this relation and then use it to construct recursively an basis of K eigenvectors of W( ) for the holonomy groups SO(n), G 2 and Spin(7). A different interpretation of the same relation between B and K is studied in the next section concerning Kähler manifolds. 4.1 The basic recursion procedure Recall that the twist τ is defined in the interpretation W(V ) = Hom g (T T V, V ) of the space of Weitzenböck formulas as linear maps T T V V by precomposition with the endomorphism τ : a b v b a v. Generalizing this precomposition we observe that W(V ) is a right module over the algebra End g (T T V ) containing τ. Interestingly both the classifying endomorphism K and the (right) multiplication by the conformal weight operator B are induced by precomposition with elements in End g (T T V ), too, say K is the precomposition with the g invariant endomorphism K : T T V T T V, a b v ν X ν a X ν b v 19

while (right) multiplication by B is precomposition with the g invariant endomorphism B : T T V T T V, a b v ν a X ν b X ν v by Fegan s Lemma 3.3. From this description of the action of the classifying endomorphism K and right multiplication of B on W(V ) we immediately conclude: K + B + τbτ = 1 2 ( CasΛ2 2Cas Λ2 T Cas Λ2 ) Applying once again Schur s Lemma this relation implies our basic Recursion Formula: Theorem 4.1 (Recursion Formula) Let be an irreducible representation of the holonomy algebra g. Then the action of K, B and τ on W( ) = Hom g (T T, ) by precomposition satisfies: K + B + τbτ = Cas Λ2 T = 2 dim h dim T We will now explain how this theorem yields a recursion formula for K eigenvectors. In fact given an eigenvector F W(V ) for the twist τ and the classifying endomorphism K with eigenvalues t and κ, i.e. τf = tf and KF = κf, the Recursion Formula allows us to produce a new τ eigenvector F with eigenvalue t. This simple prescription suffices to obtain a complete eigenbasis for W(V ) of τ and actually K eigenvectors in general Riemannian geometry g = so n and, with some modifications, also in the exceptional cases g = g 2 and g = spin 7. The quaternionic Kähler case can be dealt with similarly. Corollary 4.2 (Basic Recursion Procedure) Let F W(V ) be an eigenvector for the involution τ and the classifying endomorphism K of an ideal h g, i. e. K(F ) = κf and τ(f ) = ±F. The new Weitzenböck formula F new := ( B CasΛ2 T 2 κ ) F is again a τ eigenvector in W(V ) with τ(f new ) = F new. In particular we find: 1 new = B and B new = B 2 1 4 CasΛ2 g B. Proof: We observe that the Recursion Formula 4.1 in the form τbτ = Cas Λ2 T K B implies under the assumptions K(F ) = κf and τ(f ) = ±F that and consequently: ±τ( BF CasΛ2 T 2 ±τ( BF ) = ( Cas Λ2 T κ κ ) F BF F ) = ( BF CasΛ2 T 2 κ F ). The formulas for 1 new and B new are immediate consequences of Lemma 3.10. 20

Recall that a K eigenvector is automatically a τ eigenvector. In general however the new Weitzenböck formula F new W(V ) does not need to be an eigenvector for K again and there is no way to iterate the recursion. Nevertheless it is possible to avoid termination of the recursion procedure for most of the irreducible non symmetric holonomy algebras by using appropriate projections. We note that any +1 eigenvector of τ orthogonal to 1 is already K eigenvector in Hom g (Sym 2 0T, End ). This is due to the fact that 1 spans the other summand of the +1 eigenspace of τ. In particular the orthogonal projection of B new onto the orthogonal complement of 1, i. e. the polynomial B 2 1 4 CasΛ2 g B + 2 n CasΛ2, is a K eigenvector in Hom g (Sym 2 0T, End ). More generally we have: Corollary 4.3 (Orthogonal recursion procedure) Let p 0 (B),..., p k (B) be a sequence of polynomials obtained by applying the Gram Schmidt orthonormalization procedure to the powers 1, B, B 2,..., B k of the conformal weight operator B. If all these polynomials are τ eigenvectors and p k (B) is moreover a K eigenvector, then the orthogonal projection p k+1 (B) of B k+1 onto the orthogonal complement of the span of 1, B,..., B k is a again a τ eigenvector. Proof: Since p k (B) is a K eigenvector the basic recursion procedure shows the existence of a polynomial in B of degree k + 1, which is a τ eigenvector. It follows that with span{1, B,..., B k } = span{p 0 (B),..., p k (B)} also span{1, B,..., B k+1 } is τ invariant. Clearly the orthogonal projection p k+1 (B) of B k+1 onto the orthogonal complement of span{1, B,..., B k } is a polynomial in B of degree k + 1 with: span{1, B, B 2,..., B k } C p k+1 ( B ) = span{1, B, B 2,..., B k, B k+1 }. Now the involution τ is symmetric with respect to the scalar product on End g (T V ) and so the orthogonal complement of a τ invariant space is again τ invariant. 4.2 Computation of B eigenvalues for SO(n), G 2 and Spin(7) In this section we will compute the B eigenvalues for the holonomies SO(n), G 2 and Spin(7) by applying the explicit formula of Corollary 3.4. In particular we will see that with only two exceptions all B eigenvalues are pairwise different. This information is relevant in the proof of Proposition 3.7. The SO(n) case. Recall that in Section 2 we fixed the notation for the fundamental weights ω 1,..., ω r and the weights ±ε 1,..., ±ε r of the defining representation R n of SO(n) with r := n. Moreover the scalar product, on the dual of a maximal 2 torus was chosen so that the weights ±ε 1,..., ±ε r are an orthonormal basis. A highest weight can be written λ = λ 1 ω 1 +...+λ r ω r = µ 1 ε 1 +...+µ r ε r with integral coefficients λ 1,..., λ r 0 and coefficients µ 1,..., µ r, which are either all integral or all half integral and decreasing. Independent of the parity of n the conformal weights are b +εk = µ k k + 1, b εk = µ k n + k + 1, b 0 = r according to Corollary 3.4, where the zero weight only appears for n odd. With only a few exceptions the conformal weights are totally ordered and thus pairwise different. In the 21